# Video: AQA GCSE Mathematics Higher Tier Pack 5 β’ Paper 1 β’ Question 18

This prism has a cross section consisting of a semicircle attached to a parallelogram. All the lengths in the diagram are in centimetres. Find an expression for the total volume of the prism in cubic centimetres. Write your answer in the form π(π₯Β³ + π₯Β²) + πππ₯Β³.

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### Video Transcript

This prism has a cross section consisting of a semicircle attached to a parallelogram. All the lengths in the diagram are in centimetres. Find an expression for the total volume of the prism in cubic centimetres. Write your answer in the form π multiplied by π₯ cubed plus π₯ squared plus π multiplied by π multiplied by π₯ cubed.

Letβs begin by reminding ourselves what the formula is for the volume of a prism. Itβs the area of its cross section multiplied by its length. The cross section of this prism is a compound shape. Itβs made up of a semicircle and a parallelogram. We do know its length though. Itβs one-half π₯.

Letβs begin by working out the area of the cross section. Weβre going to calculate the area of the two shapes itβs made up of. The formula for area of a circle is ππ squared, where π is the radius. So we can find the area of a semicircle by halving this. We can find the diameter of our semicircle. Since opposite sides of a parallelogram are equal in length, this means the diameter of our semicircle is π₯. The radius is half of the length of the diameter. So itβs a half of π₯ or π₯ over two. The area of the semicircle is therefore given by π multiplied by the radius squared. Thatβs π₯ over two squared all over two.

We need to be careful when squaring π₯ over two. We have to square both the numerator and the denominator. That gives us π₯ squared over four. Having a fraction within a fraction made that last sum a little bit tricky. Instead, if we write it as a half of π multiplied by π₯ squared over four, itβs much easier to deal with.

We multiply the numerators. Thatβs one multiplied by π multiplied by π₯ squared. Thatβs ππ₯ squared. We then multiply the denominators, remembering that the denominator of π will be one. Two multiplied by one multiplied by four is eight. So the area of the semicircle is π multiplied by π₯ squared over eight or an eighth of ππ₯ squared.

Next, we need to find the area of the parallelogram. Thatβs found by multiplying its base by its perpendicular height. The base of our parallelogram is π₯ and its height is π₯ plus one. So its area is π₯ multiplied by π₯ plus one. Itβs important to include the brackets around the π₯ plus one because that reminds us that the π₯ is multiplying both the π₯ on the inside of that bracket and by the one. π₯ multiplied by π₯ is π₯ squared and π₯ multiplied by one is π₯. So the area of the parallelogram is π₯ squared plus π₯.

We can see then that the total area of the cross section is an eighth of π multiplied by π₯ squared plus another π₯ squared plus π₯. All thatβs left is to multiply this area by the length of the prism. Thatβs one-half π₯. Weβll multiply this bracket out by multiplying each term inside the bracket by one-half π₯. One-eighth multiplied by one-half is one sixteenth. So we get one sixteenth of ππ₯ cubed. π₯ squared multiplied by a half π₯ is a half π₯ cubed and π₯ multiplied by a half π₯ is a half π₯ squared.

We do need to write our answer in the form given. Notice how π₯ cubed and π₯ squared both have a coefficient of a one-half. We can therefore factorise this partially by taking out a factor of one-half. And a half π₯ cubed plus a half π₯ squared is the same as a half multiplied by π₯ cubed plus π₯ squared. And weβre left also with one sixteenth of ππ₯ cubed. Comparing this to our form given in the answer, we can see that π is equivalent to a half and π is equivalent to one sixteenth.

And weβre done! The volume of our prism is a half multiplied by π₯ cubed plus π₯ squared plus a sixteenth ππ₯ cubed.